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LINEAR CANONICAL TRANSFORMS:LINEAR CANONICAL TRANSFORMS:40 years of developments40 years of developments
Kurt Bernardo WolfInstituto de Ciencias Físicas
Universidad Nacional Autónoma de México CuernavacaCuernavaca
Beautyin Physics
Cocoyoc, 14—18 May 2012
Back in 1970…the same formulas were found at the same time in two very different places:
Stuart A. Collins Jr. Stuart A. Collins Jr. (Electroscience Lab, Ohio State University, Columbus)
described light propagation in the paraxial régime through an integral kernel, built from thin lenses and empty spaces.Lens-system diffraction integral written in terms of matrix opticsJ. Opt. Soc. Am. 60, 1168—1177 (1970)
Marcos Moshinsky Marcos Moshinsky and
Christiane Quesne Christiane Quesne (Instituto de Física, UNAM)
sought for the conservation of uncertainty under linear transformations of phase space, as a matter of intrinsicmathematical interest.Oscillator Systems. In: Proceedings of the 15th Solvay Conference in Physics (1970)Linear canonical transformations and their unitary representationJ. Math. Phys. 12, 1772-1780, 1780—1783 (1971)
Stuart A. Collins Jr.Professor Emeritus
>514 citations (Feb 2012)
The ElectroScience Laboratory
The Ohio State University
Columbus, Ohio
OptoelectronicsSpace Science
6 patents from 1982 to 2008
Lens-System Diffraction Integral Written in Terms of Matrix OpticsJ. Opt. Soc. Am. 60, 1668-1177,1970
1st
and 2nd
–order citations to the paper of Stuart Collins
citationsper year
Marcos MoshinskyInstituto de Física
Universidad Nacional Autónoma de México
Christiane QuesneFaculté
de SciencesUniversité
Libre de Bruxelles
Oscillator Systems. In: Proceedings of the 15th Solvay Conference in Physics (1970)Linear canonical transformations and their unitary representationJ. Math. Phys. 12, 1772-1780, 1780—1783 (1971)>346 citations (Feb 2012)
1st
and 2nd
–order citations to the paper of Moshinsky & Quesne
citationsper year
The paraxial optical construction with matrices(Brower 1964, Gerrard & Burch 1975)
}The Fresnel transformis integral, ergo:
1.
Find the excess quadratic phase of off‐axis rays in compound systems.2.
Find the Lagrangian
that requires
3.
Find the phase
(only
from the Fresnel transform). 4.
Find normalization
from energy conservation.
x: ray position, p: n
x
ray slope
Lens-system diffraction integral written in terms of matrix opticsJ. Opt. Soc. Am. 60, 1168—1177 (1970)
The conservation of uncertainty
under linear transformations of w = ( ),xp
Preservation of HW =>symplectic conditions.Integral transform formacting on xf and on pf =>2N
differential equations
for the kernel:for the kernel:
Limit to the identity implies
Oscillator Systems. In: Proceedings of the 15th Solvay Conference in Physics (1970)Linear canonical transformations and their unitary representationJ. Math. Phys. 12, 1772-1780, 1780—1783 (1971)
The well‐known 1‐dim LCTs
…be careful with phases:
LCTs in the Hilbert space L (R)are unitary,
They compose as Sp(2,R)‐‐up to a metaplectic sign
They include the Fourier
transform‐‐but for a phase, so they coverthe Fourier cycle twice.
2
The metaplecticmetaplectic
phasehas bedevilled countless paraxial optical papers…because their authors did not realize that Sp(2N,R)has an infiniteinfinite
cover group, of which the integral realization is the double
cover.
Valentin Bargmannhas shown thatif we use the polarmatrix decompositionwe can parametrizeany cover group (1947)
…actually, this was realized several years later…We were concerned with the harmonic oscillatorrather than with the fractional Fourier transform,and optics people do not care about overall phases.
The Lie algebra of the group of integral transformshas as generators second‐order differential operators
The
Lie algebra thatgenerates
LCTs:
Is a realization of sp(2,R)associated withparaxial optical elements
There exist Bargmann‐typeHilbert spaces ofanalytic functionswhere Complex LCT’sare unitary.
KBW, Canonical transforms I. Complex linear transforms, J. Math. Phys. 15, 1295-1301 (1974)
RadialRadial
canonical transforms
Hilbert space on R+= [0, ∞)
for m = 0, 1, 2, …
involves Bessel functions
M. Moshinsky, T.H. Seligman and KBW, Canonical radial transformations andthe radial oscillator and Coulomb problems, J. Math. Phys. 13, 901-907 (1972)
The complex extensions include Barut‐Girardello‐type integral transforms
KBW, Canonical transforms II. Complex radial transforms, J. Math. Phys. 15, 2101-2111 (1974)
HyperbolicHyperbolic
canonical transforms
Hilbert space on R+ ⊕
R+
for s e [0,∞), ϖ
e
±
involves Hankel
functions:
KBW, Canonical Transforms IV.Hyperbolic transforms:continuous series of SL(2,R)representations,J. Math. Phys. 21, 680-688 (1980)
no complex extension…
What areare
canonical transforms?Valentin Bargmann (1947), and Gel’fand and Naimark (1947 also)studied the unirreps of the 2+1 Lorentz algebra and group SO(2,1).
We have a 2nd‐order diff op realizationthat transforms under Sp(2,R) as:
The centrifugal/centripetal potential divides unirreps into discretecontinuous( and exceptional){
…and Sp(2,R) has 3 subgroup orbits
hyperbolicparabolic
elliptic(non‐exponential)
These providethe ‘row’
indicesfor matrix orintegral kernelrepresentations.
Elliptic orbit:lower‐bound orinfinite matrices
Parabolic orbitParabolic orbitintegral kernels oflinear canonical transforms
Hyperbolic orbitintegral kernels
Thus we have2
representation series in
3
subgroup reductions
Hilbert spaces undergoing unitary irreducibleHilbert spaces undergoing unitary irreducibleSp(2,R) Linear Canonical Transforms:Sp(2,R) Linear Canonical Transforms:Representationseries:
ellipticbasis
parabolicbasis
hyperbolicbasis
infinite or half‐infinitediscrete (vector) functions
functions of a radiusor pairs of the same.
functions on the real lineor pairs of the same.
‐elliptic Acts on lower‐bound infinite vectors
‐parabolic Acts on functions on a radius
LCTs on functions on the real line (Collins, Moshinsky & Quesne)
‐hyperbolic Acts on functions on the real line
‐elliptic Acts on infinite vectors
‐parabolic Acts on two‐functions on a radius
‐hyperbolic Acts on two‐functions on the real line
Thus
Sp(2,R) provides 6 faces of LCTs at least
In the parabolic basis,In the parabolic basis,The C / M‐Q LCTs belong to D(+,1/4) + D(+,3/4) .Radial LCTs belong to the discrete series D(+,k) for k > 0 .Hyperbolic LCTs belong to continuous series C(e,s) for k = 1/2 + i s .In the elliptic basiswe can have lower‐bound infinite matrix LCTs in the discrete series,or infinite matrix LCTs in the continuous series.
The hyperbolic basis for 2‐component functions remains unused.The exceptional interval 0 < k < 1 has 1‐param self‐adjoint extensions.
The exceptional representation series is terra incognita (hic vivunt leones!).
Conserveduncertaintyrelations for `classical’
LCTs
There has been much recent interest
in FINITE LCTs to permit procesing by computers
But of course…
non‐compact groups do not have
unitary finite‐dimensional faithful representations !!!
Preferably using the FFT algorithm for
So, people use:
A ‘finite fractional Fourier transform’
presents multi‐solutions,and a toroidal phase space will not rotate continuously.
Finite data sets divide us into two cities:in one, those who believe in symmetries,in the other, those who want fast results.
Use discretized
version of LCTsnot
unitary, do not composetoroidal
phase space a mess…
Use the group SU(2).Enjoy unitarity, composition,and a nice
spherical phase space.
¡ Muchas gracias!
PAPIITPAPIIT--UNAMUNAMand SEPSEP--CONACYTCONACYT
projectsÓptica Matemática
¡ Muchas gracias!
PAPIITPAPIIT--UNAMUNAMand SEPSEP--CONACYTCONACYT
projectsÓptica Matemática
With
best admiration
and
wishesfor
Prof. Francesco Iachello
on
his
70 years
of
success
!
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